In a large pot, bring the chicken broth to a simmer over medium heat (do not let it boil). Add the chicken, cover and lower the heat to medium-low. Simmer until the chicken is just cooked through, 15 to 20 minutes. Using a slotted spoon or tongs, transfer the chicken to a bowl and set aside to cool.

Return the broth to a simmer. Add the onion, garlic, black pepper, and mixed vegetables, cover and cook until the vegetables just begin to soften, about 15 minutes. Add the bell pepper and jalapeño (if using), cover, and simmer until the mixed vegetables are tender enough that the outsides can be pierced with a fork but there is still some resistance in their centers, 10 to 20 minutes.

Shred the chicken. Raise the heat to medium, add the chicken, hominy, and matzo balls to the soup, and bring it to a gentle simmer. Lower the heat to low, cover, and cook for 20 minutes. Add one cup of the chopped tomatoes, the corn, and the spinach and simmer until the spinach is wilted and the matzo balls are heated through, about another five minutes. Add the salt. Taste and add more salt and black pepper, if desired.

Serve topped with the remaining 1/2 cup chopped tomatoes, the cilantro, the green onions, and radishes or place the garnishes in individual bowls and pass them at the table so diners can make their choice. Squeeze lemon juice into the soup and stir before eating. Pass the salsa to stir in (if using).